BOOTSTRAP CONFIDENCE BANnS FOR THE RENEWAL FUNCfION
by
John Crowell and Pranab K. Sen
Western Michigan University
and
University of North Carolina
Let F be a lifetime distribution and H the associated renewal function.
Two
nonparametric estimators of H are shown to have desirable asymptotic properties
without moment restrictions on F.
One of these estimators. the empirical
renewal function. is then used to construct an asymptotically correct bootstrap
confidence band for H on a finite interval.
Weak convergence of the
bootstrapped empirical renewal function is established using weak convergence
of the bootstrapped empirical process.
AMS 1980 subject classifications. Primary 6OK05; secondary 62G15.
Key words and phrases.
weak convergence.
Abbreviated title:
Renewal function estimation. bootstrap confidence band.
Renewal Function Confidence Bands
1.
Introduction
Let X, Xl' X , ... be independent and identically distributed nonnegative
2
random variables with distribution function F.
F(k)(t)
= P(~ ~
With SK
= Xl
+ ... + X and
k
t) (the k-th convolution of F), the renewal function H is
defined by
00
=
H(t)
!
F(k)(t),
t ~ O.
k=l
The renewal function occurs in a number of different applications, including
some where F is not a lifetime distribution (See Frees (1986b».
Here
attention is restricted to nonnegative random variables, this being the most
important special case for applications.
The principal objective of this paper
is constructing a confidence band for H on a finite interval when F is a
lifetime distribution.
For large values of t, the renewal function can be
estimated by exploiting an asymptotic expression for H(t) as t
~
00,
as
described in Frees (1986a).
Several recent papers have considered nonparametric estimation of H using
the random sample Xl" .. ,X .
n
Let I(E) be the indicator of the event E.
An
estimator introduced by Frees (1986a.b) is of the form
"-
Hn (t)
(1.1 )
where m
~
n and m
~ 00
as n
~ 00.
Here
is the U-statistic for F(k)(t), the summation being over all subsamples of size
e-
- 2 -
k taken without replacement from {X1 •...• X }.
n
Frees establishes strong
~
~
A
consistency of H (t) and asymptotic normality of n [H (t)-H{t)] for fixed t
n
n
under several sets of conditions on m and the moments of F.
n-
1
~=l
r{X
i
~ x).
-00
< X < 00.
Let F (x)
n
~
0
=
be the usual sample distribution function and
F{k) denote its k-th convolution.
n
Frees (19S6b) also discusses the estimator
H (t)
(1.2)
n
which is just the renewal function that arises from sampling with replacement
from F .
n
Schneider, Lin, and O'Cinneide (1990) suggest algorithms for computing
both (l.l) and (1.2), as well as comparing the performance of the two
estimators on data and in a simulation study.
They conclude that H may often
n
be the preferred estimator because the order of computation is less, although
H may fail to perform well when F has rapidly decreasing failure rate.
n
Let T be a finite positive constant and D[O.T] be the space of functions
on [O,T] that are right-continuous and have left-hand limits.
Schneider (1988) show that the process r
n
= {n~
O'Cinneide and
[Hn{t) - H{t)], t E [O,T]}
converges weakly to a Gaussian process in the Skorohod topology on D[O,T] when
F is a continuous lifetime distribution.
conditions the process r
n
We prove that under the same
can be bootstrapped to construct an asymptotically
correct confidence band for H on the interval [O.T].
In Section 2 we note that when F is a lifetime distribution no moment
conditions are necessary for strong consistency and asymptotic normality of H ;
n
the same properties then hold for H.
n
The bootstrap procedure is defined in
Section 3; Theorem 3.1 gives conditions for the asymptotic validity of the
confidence band.
Section 4 contains a proof of weak convergence for the
- 3 -
bootstrapped version of the process r n .
2.
"
Asymptotic Properties of Hn(t) and Hn(t)
Theorems 2.2 and 3.1 of Frees (l986b) state the strong consistency and
"
asymptotic normality of H (t) under several sets of conditions on the moments
n
of F and the design parameter m; an underlying assumption is that F has
positive mean ~ and finite variance
0
2
In the present section it is shown
.
that no moment assumptions are necessary when F(O-)
indicated in Theorem 2.1.
t €
< 1.
and F(O)
as
The asymptotic properties then carryover to H
n
without further restrictions on F.
Theorem 2.1:
=0
= O.
Assume F(O-)
(See Theorem 2.2.)
F(O)
< 1.
and log n
= oem).
Then for each
IR
"
Hn (t)
(2.1)
~
e-
H(t) a.s.
and
(2.2)
where
}:
r.s=1
PROOF:
Assuming that F has positive mean
~
and finite variance
0
2
,
a
sufficient condition for (2.1) and (2.2) given by Frees (1986b) is that
(2.3) for some a 1
Here X-
>0
= min(O.X).
and all lal
Let p
=f
o < ~ < 00 is used to show that
e
-u
< aI'
E exp(-aX-)
dF(u).
<
00
and log n
= oem).
Assuming (2.3), the condition that
- 4 -
<p <1
(2.4) 0
and there exists a 9
2
for all k
~
I, t €
>0
such that F(k}(t} ~ exp(9 t} pk
2
m.
Frees' proofs of strong consistency and asymptotic normality assuming (2.3)
rely solely on the inequality (2.4) and the condition that log n
this case, the assumption of a finite variance can be dropped.
= oem}:
in
If F(O-}
=0
= I,
and F(O}
< I,
EX =
Taking log n = o(m}, (2.l) and (2.2) follow by the same arguments used
00.
then it is easily shown that (2.4) holds with 9
by Frees assuming (2.4) holds and log n
2
even when
= oem}.
o
Frees (1986b) noted that by exploiting the relationship between the
U-statistic F(k}(t} and the V-statistic F(k}(t} for k
n
n
= 1, ... ,m.
asymptotic
A
properties of H (t) could be shown to hold for a truncated version of H under
n
certain conditions on F.
n
Here we use Theorem 2.1 to establish the asymptotic
behaviour of H when F is a lifetime distribution.
n
Theorem 2.2.
=0
Assume F(O-}
and F(O}
H (t)
(2.5)
n
~
< 1.
Then for each t €
m
H(t) a.s.
and
~
n [Hn (t) - H(t}]
(2.6)
D
~
2
N(O,a t ).
2
where at is defined in Theorem 2.1.
PROOF:
hold.
(2.7)
6
Take m such that m
We first show that
= o(n)
and log n
= oem}.
Then (2.1) and (2.2)
- 5 -
and
Ol)
n
(2.8)
Yz
1:
k=m+1
k
Since n -n(n-1)(n-2) ... (n-k+1)
~
_k
2j k-j
rj=1 k
n
• it follows that for any
positive integer r and a positive constant C
Here C depends on r but not on n.
> O.
Thus by the Minkowski inequality
(2.9)
(See Serfling (1980).)
Taking r sufficiently large. by applying the Markov
inequality and Borel-cantelli lemma. (2.7) holds.
Next let p
take 8
(2.10)
holds.
= f e-u dF (u).
n
Applying (2.4) to the distribution F
n
(we may
= 1 for any lifetime distribution). we see that
2
Since p
n
n
n
~
Yz
p a.s. by the strong law of large numbers and log n = oem). (2.8)
Thus we have (2.5).
Using (2.9) and the Markov inequality. it can be shown that
which with (2.8) implies (2.6).
3.
o
A Bootstrap Confidence Band
In the remainder of this paper. the product of two functions without their
arguments will denote convolution.
For example.
~(t) = f~ F(t-u) dF(u) =
- 6 -
The symbol
W
~
will represent weak convergence. where unless
otherwise specified this is understood to be in the Skorohod topology on
D[O.T].
For any function f. let IIfll = sup
If(t)l.
tE[O.T]
o
Let W be a Brownian bridge on [0.1] and G(t) =
where F(O)(t) = I(t ~ 0).
n
~ (B
0
(k+l)F
As in the introduction. define r
{n~ [Hn (t) - H(t)]. t E [O.T]}.
that r
CD
~=O
n
(k)
(t). t
~
O.
=
O'Cinneide and Schneider (1988) have shown
F)G when F is continuous.
Here
denotes composition of
0
functions.
To construct a confidence band for H on the interval [O.T]. one would like
some knowledge of the distribution
J(x) = P{II(B
0
F)GII
~
x}.
-CD
< X < CD.
If the desired level of confidence was I-a and one knew
~
c = inf{x : J(x)
I-a}. then {Hn(t)
±n
.JA
c. t E [O.T]} would be a reasonable
large sample confidence band for H having the correct asymptotic coverage
probability when J is continuous at c.
the process r
n
We now describe how by bootstrapping
we may estimate J and construct an asymptotically correct
confidence band for H on [O.T].
Let q =
and B
~ CD
~
as n
and B = B be sequences of positive integers such that q
n
~ CD.
~ CD
Given the random sample Xl ..... X . let Xl* ..... X* be drawn
n
q
* .... X*q are conditionally
with replacement from {Xl" ..• Xn }: that is, Xl'
independent with distribution F .
n
The empirical distribution function of the
q
bootstrap sample is then Fnq(t) = q -1 Ii=l
I(X*i
convolution of F
~
~
t).
Let F(k) be the k-th
nq
Define the bootstrapped version of the process r
r: =
{q~[H:(t) - Hn(t)].
t E [O,T]}
n
by
- 7 -
*
(k)
0)
In Section 4 it is shown that if F is a continuous
where Hn(t) = ~=l Fnq (t).
lifetime distribution. then the conditional distribution of r n* given Xl •.... Xn
converges weakly to (B
F)G along almost all sample sequences.
0
Given Xl •.... X . take r * .. ·· .r* to be conditionally independent copies
n
nB
nl
of r*.
n
Define
J (x)
n
and take c
n
= inf{x
= B- 1
B
I(lIr* .11
};
nJ
j=l
~
In(x)
I-a}.
{H (t) ± n
~
x).
-Q)
<x <
0).
A confidence band for H is then given by
-~
n
c .
n
t €
[O,T]}.
The following theorem confirms that the coverage of the confidence band is
2
asymptotically correct when q = o(n ).
Theorem 3.1:
Assume F is absolutely continuous. F(O) = O. and F(T)
e-
> O.
2
If q = o(n ). then
lim
P{ IIHn - HII
n,q.JHn
~
n -~ c n IXl .... ' Xn } = I-a
along almost all sample sequences.
PROOF:
By Theorem 4.1. J (x)
n
~
J(x) for every continuity point x of J.
If J
is continuous at c. then
lim
P{IIHn - HII
n.q.JHn
along almost all sample sequences.
~
n -~ Cn IXl ..... X}
= I-a
n
When F is continuous and F(T)
> O.
the
distribution J cannot have an atom at zero and is necessarily continuous. which
completes the proof.
(See Csorg(f and Mason (1990).)
o
- 8 -
= o(n2 )
The assumption that q
is not restrictive since in practice one
= O(n).
typically takes q
Theorem 3.1 allows one to test the null' hypothesis that H(t)
t € [O,T], where H is some user specified function.
o
-~
reject the null hypothesis when IIH -H II > n
c.
non
would have size
= Ho (t),
Specifically, one would
Asymptotically this test
under the conditions of Theorem 3.1.
a
~
More typically one would like to test that F belongs to some class
of
o
distribution functions, and hence H belongs to the corresponding class of
renewal functions
~
o
Alternatively, one could test directly that H
.
if for somme H' €
the following test:
the null hypothesis that H
€ ~.
exponential distributions with positive mean
{H : H(t)
= t/~,
t
~
retained if for some
0, ~
~
> O}.
> 0,
~
>0
~
n
using
were the class of
0
so that
~
o
Then the null hypothesis H €
IIH (t) -
o
~o one has IIHn-H'II <
n-~ c n , then retain
-
For example, suppose
o
€ ~
t/~II ~
n
-~
=
~
o
would be
c, a seemingly reasonable test
n
in this case.
4.
Weak Convergence
Theorem 4.1:
F(O)
= O.
Assume F is an absolutely continuous distribution such that
If q
= o(n2 ),
then the conditional distribution of r * given
n
Xl' ... ,X converges weakly to (B
n
0
F)G as n, q
~
00
along almost all sample
sequences.
PROOF:
Throughout this section let U(x) = F(O)(x) = I(x
~
0).
Expressions of
the form U/(U-F) will represent power series, with convolution taking the place
of multiplication.
For example,
U
U-F (t)
=1
+ H( t) ,
t ~
O.
- 9 -
O'Cinneide and Schneider (1988) exploited an identity for rn(t) to show
We utilize the same identity for r * • namely
n
weak convergence of r .
n
q~[H* - H ]
( 4.1)
n
n
= q~[F
+
nq
- F ]
U
n (U-F)2
n
1 [~(F
~ q
q
nq -
F )]2
n
U
U
U-F
(U-F)2'
~
n
Adapting the approach used by O'Cinneide and Schneider (1988). we first show
that for any c
>0
1
U
P{II[q~ (F -F)] 2
~
nq n
U-F
(4.2)
U
nq (U-F)2
n
q
-+ 0
as
n ,q -+
II
> c IXl.··· ,Xn }
(I).
By Theorem 2.2
II
(4.3)
Similarly, IIU/(U-F
nq
)11
U~F
~
n
II
~
1 + Hn(T) -+ 1 + H(T)
a.s.
*
1 + Hn(T);
we show that there exists a positive
constant M such that
(4.4)
By a well known probability inequality (cf. Serfling (1980». for any c
P{IIF -FIl
nq
(4.5)
Let pn.....~
=I
>c I
X1 •...• X } -+0
n
>0
n,q -+(1).
e-u dFnq (u) and as in Section 2 (using (2.4»
(4.6)
Choose 0
as
such that F(o) ~ e- o/ 2 .
Then
>0
~-
- 10 -
(4.7)
and by (4.5), (4.6) and (4.7), we have that (4.4) holds.
Bickel and Freedman (1981) showed that for almost all sample sequences the
conditional distribution of the bootstrapped empirical process q~[F -F]
nq n
converges weakly to B
Letting G (t)
n
0
F.
= -kLoo=o
Thus (4.2) follows from (4.3) and (4.4).
(k+1)F(k)t, where F(O)(t)
n
n
= F(O)(t) = U(t),
we can
write
= q~ [Fnq
- Fn ]G + q~ [Fnq - Fn ] [Gn - G],
where G was defined in Section 3.
As noted by O'Cinneide and Schneider (1988),
when F is continuous G is a continuous, monotone increasing function on [O,T].
Thus convolution with G is a continuous mapping of D[O,T] into D[O,T].
Using
Theorem 4.1 of Bickel and Freedman (1981) and Theorem 5.1 of Billingsley
W (B
0
F)G for almost all sample sequences.
It remains to be shown that for any
~
>0
~
~
(1968), it follows that q [Fnq-F]G
n
P{lIq~[Fnq-Fn ][Gn-G]II > ~
(4.8)
I Xl' ... ,Xn }
~
0
as
n,q ~
00.
As pointed out in Section 2,
p
=f
n
e-u dF (u) ~ p
n
=f
e-
u
dF(u)
a.s.
Thus using (2.4) and dominated convergence,
G (t)
(4.9)
n
~
G(t)
a.s.
for all t € [O,T].
We next define continuous versions of the empirical functions F and F
n
nq
Let t
1
< t 2 < ... < t n
be the points of discontinuity of F.
Wi th t
o
= 0,
- 11 -
define
= {Fn(t i )
1,
+ n
t
~
-1
t
n
(t-t i )/(t i + 1-t i ). t i ~ t
< t i +1 .
i
= O•...• n-1;
.
Similarly define F~q by linearly interpolating between the successive
discontinuities of Fnq
Since F is continuous and q
= o(n2 ).
~
IIq~[Fo
n - Fn ]11 ~ 9.....
n
(4.10)
and for any
t
>0
0
.
it can be shown that
>t I
0 ]11
P{lIq~ [Fnq-Fnq
(4.11)
-+
X •...• X} -+0.
n
1
Thus
~
q [F
( 4.12)
0
nq
0
- F ]
n
W
-+
B
0
F
a.s ..
where this weak convergence takes place in the uniform topology on C[O.T]. the
space of continuous functions on [O,T].
By (4.9) and (4.10).
~
( 4.13)
IIq~[Fo-F
] [G -G]II ~ 9..... [G (T) + G(T)] -+
n n
n
n
n
0
as
Similarly employing (4.9) and (4.11) yields that for any
o
P{lIq~[Fnq _Fnq
] [Gn -G]II > t I
( 4.14)
n. q -+
t
>0
Xl.··· .X } -+ O.
n
Thus we would like to establish that
P{lIq~[Fo
nq -Fo][G
n
n -G]II > t I
(4.15)
as n. q -+
00.
Xl.·· .• X } -+ 0
n
00.
~.
- 12 -
Let x € C[O.T] and {x } any sequence in C[O.T] such that x
n
uniform metric.
n
Then given Xl •...• X • x [G -G]
n
n
n
~
~
x in the
0 in the Skorohod metric on
D[O.T]. where 0 is the function which is identically zero on [O.T].
Thus by
(4.l2) and Theorem 5.5 of Billingsley (1968). given Xl.··· ,Xn
in the Skorohod topology on D[O.T].
Thus (4.15) holds. and by (4.13). (4.14).
and (4.15) the desired conclusion (4.8) holds as well.
o
- 13 -
REFERENCES
Bickel. P.J. and Freedman. D.A. (1981).
bootstrap.
Some asymptotic theory for the
Ann. Statist. 2. 1196-1217.
Billingsley. P. (1968).
Convergence of Probabitity Measures.
Csorg~ S. and Mason. D.M. (1990).
Wiley. New York.
Bootstrapping empirical functions.
Ann.
Statist. 17. 1447-1471.
Frees. E.W. (1986a)
Warranty analysis and renewal function estimation.
Nauat
Res. Logist. Quart. 33. 361-372.
Frees. E.W. (1986b).
Nonparametric renewal function estimation.
Ann. Statist.
14. 1366-1378.
O·Cinneide. C. and Schneider. H. (1988).
renewal function.
Working paper.
Weak convergence of the sample
Department of Mathematics. University
of Arkansas. Fayetteville.
Schneider. H.. Lin. B. and O·Cinneide. C. (1990).
estimators for the renewal function.
Serfling. R. (1980).
Comparison of nonparametric
Appt. Statist. 39. 55-61.
Approximation Theorems of Mathematicat Statistics.
Wiley. New York.
DEPARTMENT OF MATHEMATICS & STATISTICS
WFSfERN MICHIGAN UNIVERSITY
KALAMAZOO. MI 49008
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
CHAPEL HILL, NC 27599-3260
4It'